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Nonparametric Detection and Estimation of Highly Oscillatory Signals


Helgason, Hannes (2008) Nonparametric Detection and Estimation of Highly Oscillatory Signals. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/SAEK-MV13.


This thesis considers the problem of detecting and estimating highly oscillatory signals from noisy measurements. These signals are often referred to as chirps in the literature; they are found everywhere in nature, and frequently arise in scientific and engineering problems. Mathematically, they can be written in the general form A(t) exp(ilambda varphi(t)), where lambda is a large constant base frequency, the phase varphi(t) is time-varying, and the envelope A(t) is slowly varying. Given a sequence of noisy measurements, we study two problems seperately: 1) the problem of testing whether or not there is a chirp hidden in the noisy data, and 2) the problem of estimating this chirp from the data.

This thesis introduces novel, flexible and practical strategies for addressing these important nonparametric statistical problems. The main idea is to calculate correlations of the data with a rich family of local templates in a first step, the multiscale chirplets, and in a second step, search for meaningful aggregations or chains of chirplets which provide a good global fit to the data. From a physical viewpoint, these chains correspond to realistic signals since they model arbitrary chirps. From an algorithmic viewpoint, these chains are identified as paths in a convenient graph. The key point is that this important underlying graph structure allows to unleash very effective algorithms such as network flow algorithms for finding those chains which optimize a near optimal trade-off between goodness of fit and complexity.

Our estimation procedures provide provably near optimal performance over a wide range of chirps and numerical experiments show that both our detection and estimation procedures perform exceptionally well over a broad class of chirps. This thesis also introduces general strategies for extracting signals of unknown duration in long streams of data when we have no idea where these signals may be. The approach is leveraging testing methods designed to detect the presence of signals with known time support.

Underlying our methods is a general abstraction which postulates an abstract statistical problem of detecting paths in graphs which have random variables attached to their vertices. The formulation of this problem was inspired by our chirp detection methods and is of great independent interest.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:ChirpLab; chirplet chain; chirplet graph; chirplet path; chirplets; chirps; detection of sinusoid of unknown support; dynamic programming; FM modulation; GLRT; graphs; gravitational waves; LIGO; network flow algorithms; noise removal; nonparametric detection; nonparametric estimation; optimal estimation; optimal test; path detection in graphs; signal processing; unknown support
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Awards:The W.P. Carey and Co., Inc., Prize in Applied Mathematics, 2008.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Candes, Emmanuel J.
Thesis Committee:
  • Candes, Emmanuel J. (chair)
  • Weinstein, Alan Jay
  • Lorden, Gary A.
  • Owhadi, Houman
  • Abu-Mostafa, Yaser S.
Defense Date:30 August 2007
Non-Caltech Author Email:hannes.helgason (AT)
Record Number:CaltechETD:etd-05112008-152328
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1726
Deposited By: Imported from ETD-db
Deposited On:09 Jun 2008
Last Modified:17 Jan 2020 22:10

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